NUMERICAL ANALYSIS OF THE BYPASS VALVE IN … ANALYSIS OF THE BYPASS VALVE IN A LOOP HEAT PIPE Michel Speetjens & Camilo Rindt Laboratory for Energy …

NUMERICAL ANALYSIS OF THE BYPASS VALVEIN A LOOP HEAT PIPE

Michel Speetjens & Camilo Rindt

Laboratory for Energy TechnologyMechanical Engineering DepartmentEindhoven University of Technology

The Netherlands

Mai 2007

MATEO-ANTASME deliverable 9.2

Introduction

The present study concerns a follow-up study on the integral thermodynamical analysis of thebypass valve discussed in Speetjens & Rindt (2006). This bypass valve is a crucial componentin the loop heat pipe (LHP) that, in turn, is part of the cryo-cooling system for the thermalcontrol of the Alpha Magnetic Spectrometer. The principal purpose of the bypass valve isaverting freezing of the working fluid (Propylene) – and thus malfunctioning of the LHP – bytermination of its circulation through the LHP in case the evaporator pressure drops below apreset minimum value. Further background and details are in Speetjens & Rindt (2006).

The integral analysis of the bypass valve in Speetjens & Rindt (2006) leans on the assump-tion of (i) inviscid flow, (ii) uniform inlet/outlet conditions and (iii) quasi-static conditions.The present analysis seeks to determine whether viscosity, non-uniformity and non-quasi-static effects (e.g. due to fluid inertia) are significant, which requires resolution of the fluidflow and heat transfer in the interior of the bypass valve. This analysis is performed throughnumerical simulation of the two steady-state operating conditions of the bypass valve by meansof the finite-volume method (FVM). To this end we use the commercial package Fluent.

Operating conditions and material properties

The present analysis is, as opposed to the non-dimensional study in Speetjens & Rindt (2006),performed in dimensional quantities so as to specifically characterise the actual bypass valve.The geometry and corresponding dimensions of the bypass valve and the operating conditionsare specified in Bodendiek et al. (2005):

• geometry as shown schematically in Figure 1a (dimensions as in Bodendiek et al.(2005));

• temperature range at the inlet: 245K ≤ T ≤ 265K;

• saturation conditions at the inlet;

• negligible radiative heat loss through the walls;

• heat intake by the evaporator Qe ∼ O(30 W ); full liquid-to-vapour phase change inevaporator: Qe = mhfl, with m the mass flow and hfl the evaporation enthalpy of thePropylene;

• zero-gravity conditions.

The working medium of the LHP is Propylene (C3H6). The corresponding thermodynamicaland material inlet properties are (ALLPROPS):

p245 = 2.3bar, ρ245 = 6.0kg

m3, hfl,245 = 416.4

kJ

kgK, cp,245 = 1426

J

kgK, µ245 = 15.7µPas,

p265 = 4.6bar, ρ265 = 9.7kg

m3, hfl,265 = 416.4

kJ

kgK, cp,265 = 1568

J

kgK, µ265 = 16.3µPas,

completed by c245 = 227.5m/s and c265 = 228.1m/s as propagation speeds of sound, with thesubscripts referring to the inlet temperature. Note that, somewhat counter-intuitively, densityand kinematic viscosity increase with temperature due to the fact that, given saturation

conditions are maintained, the pressure also increases. The above leads to the followingoperating conditions at the inlet:

m =Qe

hfl∼ O(50µg/s), U =

m

ρA∼ O(3m/s), Re =

Udρ

µ∼ O(3000), Ma =

U

c∼ O(0.015)

with A = πd2/4 the inlet area and di = 2mm the inlet diameter and Re and Ma the well-known Reynolds and Mach numbers, respectively.

Mathematical model for the fluid flow and heat transfer in thebypass valve

The assumptions underlying the analysis by Speetjens & Rindt (2006) admit description ofthe fluid flow and heat transfer in the bypass valve by the integral conservation laws formass and energy of classical thermodynamics (Cengel & Boles (2002)). Dismissal of theassumption of quasi-static conditions requires this model be extended by the conservationlaw for momentum. Furthermore, allowing for viscous effects and non-uniform inlet/outletconditions introduces viscous terms in the momentum and energy equations and impliesthat specific boundary conditions on both inlet and outlet as well as on stationary wallsbecome significant and resolution of the problem within the interior of the bypass valvebecomes necessary. The model that describes the present problem thus consist of the genericconservation laws for mass, momentum and energy according to Hirsch (1988) complementedby suitable boundary conditions. However, the operating conditions in the bypass valve admitsimplification of this generic model.

The fluid in question (Propylene) is gaseous and, inherently, we in principle have a com-pressible and non-isothermal flow. However, the following considerations admit reductionto a fully incompressible and isothermal flow that can be tackled with standard numericaltechniques:

1. The present Reynolds number approximately coincides with the lower end of the tran-sition regime 2300 < Re < 104. This basically implies laminar flow conditions (VDIWarmeatlas (2002)).

2. The present Mach number is well below unity. This to good approximation impliesincompressible flow and, consequently, a density that is dependent upon the temperatureonly: ρ = ρ(T ) (Schlichting & Gersten (2000)).

3. The integral thermodynamical analysis revealed that adiabatic conditions (i.e. negligi-ble heat loss through the walls) lead to nearly iso-thermal conditions, suggesting thatinternal temperature variations are small (Speetjens & Rindt (2006)). This, combinedwith the small length scales involved with the bypass valve, admits further reductionof the simplified flow problem to the well-known Boussinesq approximation, in whichdensity variations due to thermal effects are restricted to an additional gravitationalterm in the momentum equation (Kundu (1990)). However, the bypass valve operatesunder zero-gravity conditions, meaning said gravitational term drops out here.

The above considerations simplify the original flow to an incompressible and isothermalflow that is governed by

∇ · u = 0, ρu · ∇u = −∇p + µ∇2u, ρcpu · ∇T = λ∇2T + φ, (1)

consisting of the standard Navier-Stokes equations and the energy equation, with φ repre-senting viscous heat production.

Relations (1) constitute the model that, complemented by suitable boundary conditions,describes the fluid flow and heat transfer in the bypass valve. The flow boundary conditionsare: prescribed mass flow at inlet; outflow conditions at the outlet; no-slip conditions atthe interior walls. The thermal boundary conditions are: inlet conditions as specified above;outflow conditions at the outlet; adiabatic interior walls. The laminar flow conditions allowapplication of standard numerical techniques without the need for turbulence models.

Numerical analysis of the bypass valve

Relevant characteristics for practical purposes are total pressure drop and temperature changesas a function of inlet conditions. Further relevant property is occurrence of high-pressure zoneswithin the interior of the bypass valve, which may serve as indicators for the risk of internalcondensation. The numerical analysis is restricted to the states at minimum (T = 245K)and maximum (T = 265K) inlet temperatures for both operating modes. These states areexamined for the mass-flow range 50µg/s ≤ m ≤ 100µg/s and are considered representativeof states at intermediate 245K < T < 265K.

Figure 1a shows the cross-section of the bypass valve (panel a) and the correspondingcomputational meshes of the closed (panel b) and open (panel c) operating modes. (Note thatfor legibility a coarser mesh than that actually used is shown.) The bypass valve is symmetricabout the plane spanned by the centrelines of the inlet/outlet pipes and the centreline of thebypass valve. The three-dimensional internal flow and temperature field are also assumedsymmetric about this plane. This symmetry admits reduction of the computational domainto one of the two subdomains divided by this plane of symmetry. Thus shown meshes concernonly one such subdomain.

Numerical simulations of both operating modes reveal that the bypass valve approaches anisothermal state for the operating ranges considered here. Deviations ∆T of the internal tem-peratures deviate from the prescribed inlet temperature Ti are typically ∆T ∼ O(0.002K).This signifies an essentially isothermal state T (x) = Ti at any position x within the by-pass valve. This implies that viscous heat generation is negligible and thus consolidates theassumption of an isothermal flow.

Fluid flow and heat transfer: closed mode

Figure 2 gives the typical velocity field in the closed mode. Shown are the velocity vec-tors (panel a) and the velocity magnitude (panel b), with color-coding as indicated, in thesymmetry plane for m = 70µg/s and T = 245K. The plots clearly reveal the fluid flowingdirectly from the inlet pipe via the lower-right part of the stem to the outlet. Fluid motionin other regions is negligible. The organised and confined flow structure and confinement offluid motion to said regions signifies in essence laminar flow conditions.

Figures 3a and b give close-ups of the velocity vectors and velocity magnitude, respec-tively, in the region with significant fluid flow. The entering flow has a developed profile thatis deflected downward by the stem towards the inlet of the outlet pipe. The fact that belowthe stem the velocity magnitude remains at values comparable to that in the inlet and outletpipes, despite considerable narrowing of the throughflow area, implies spreading of the fluid

underneath the stem in lateral direction (i.e. away from shown symmetry plane) after leav-ing the inlet and subsequent concentration upon reaching the inlet of the outlet pipe. Theperpendicular orientation of the outlet pipe relative to the incoming flow causes a suddenredirection of the fluid motion and leads to an asymmetric velocity profile within the outletpipe. This asymmetry persists throughout the outlet pipe and thus means that its length isinsufficient for the flow to fully develop.

Figures 3c and d give close-ups of the velocity vectors and velocity magnitude, respectively,in the abovementioned region for the same mass flux (m = 70µg/s) yet at temperatureT = 265K. The plots show that differences with the case T = 245K are marginal. The flowstructure is virtually identical; only its magnitude is slightly smaller. The latter is a directconsequence of the fact that ρ265/ρ245 > 1, which for identical m implies U265/U245 < 1. Thesimilarity in structure with small changes in conditions is a further indication of the essentiallylaminar nature of the flow. This substantiates the assumption of laminar flow conditions thatunderly the simplified flow model and corresponding numerical methods used here.

Typical pressure fields for the closed mode equal that shown in Figure 4 for m = 70µg/s.Top and bottom rows correspond with T = 245 and T = 265, respectively; left and rightcolumns correspond with full views and close-ups, respectively. The full view in panel aclearly reveals the approximately linear pressure gradient in the inlet pipe (consistent withthe developed velocity field) and the pressure build-up at the right side of the stem due todeflection of the fluid. The close-up in panel b exposes weaker pressure build-up regions onthe valve wall directly below the high-pressure region at the stem and at the left side of theinlet of the outlet pipe. The latter results from the redirection of the flow mentioned before.Panel b furthermore shows that the pressure distribution within the outlet pipe is equallyasymmetric as the velocity field. The latter leads to a weak pressure gradient transverseto the flow direction. Figures 4c and d show full view and close-up of the pressure field,respectively, for T = 265. The pressure field for T = 265 is, similar as for the velocity field, ofessentially the same structure as that for T = 245, albeit with weaker magnitude. Pressurebuild-up regions occur at the same locations yet are less profound.

Important for the thermodynamical state of the fluid is that in the high-pressure zoneat the right side of the stem the local pressure exceeds the inlet pressure. Moreover, thepressure excess at the high-pressure zone increases with increasing mass flow m. This impliesthat, given saturation conditions at the inlet and an isothermal state (see before), locallyT = Tsat and p > psat (‘sat’ denotes saturation) and thus a local pressure above the saturationpressure. Thus the simulations, though concerning incompressible flow, strongly suggest thatcondensation (or even freezing) may occur in the direct proximity of the right side of thestem due to compression of a saturated fluid (Cengel & Boles (2002)). Such phase changesmust be avoid, as they are likely to obstruct throughflow by narrowing the flow passage ormay even lead to full blockage, thus seriously compromising the performance of the bypassvalve and, inherently, the LHP. However, the above reveals that flow structure and qualitativethermodynamical state are essentially the same for slight variations in inlet conditions. Thuscondensation (or freezing) can be avoided by ensuring a slightly superheated state at the inlet.This “safety superheat” must increase with increasing m.

Fluid flow and heat transfer: open mode

Figure 5 give the global velocity field in the symmetry plane for m = 70µg/s and T = 265Kfor open-mode conditions. Here the fluid must, contrary to the closed mode, flow fully around

a) Bypass valve b) Closed mode c) Open mode

Figure 1: The bypass valve and the corresponding computational meshes for the closed andopen operating modes. For legibility a coarser mesh than that actually used is shown.

Z

Y

X Z

Y

X

a) Velocity vectors. b) Velocity magnitude.

Figure 2: Velocity field in the symmetry plane for the closed mode at m = 70µg/s andT = 245K. The velocity magnitude ranges from 0m/s (blue) to 5m/s (red).

the stem and the corresponding shaft for reaching the outlet. This leads to a distributionof inflowing fluid around the entire stem and shaft and, by virtue of mass conservation, toa significant deceleration in flow velocity in the interior of the valve. The flow becomesdirectional and fluid again acquires speed upon reaching the inlet of the outlet pipe. This

Z

Y

X Z

Y

X

a) Velocity vectors (T = 245 K). b) Velocity magnitude (T = 265 K).

Z

Y

X Z

Y

X

c) Velocity vectors (T = 265 K). d) Velocity magnitude (T = 265 K).

Figure 3: Close-ups of the velocity field in the symmetry plane in closed mode at m = 70µg/sfor T = 245K and T = 265K. The velocity magnitude ranges from 0m/s (blue) to 5m/s(red).

flow behaviour is clearly visible in shown graphs.Close-ups of the velocity field at the inlet and outlet sections are given in Figure 6. Flow

Z

Y

X Z

Y

X

a) T = 245K; full view. b) T = 245K; close-up.

Z

Y

X Z

Y

X

c) T = 265K; symmetry plane. d) T = 265K; close-up

Figure 4: Pressure field in the symmetry plane in closed mode at m = 70µg/s for T = 245Kand T = 265K. The pressure range is 135Pa; blue and red indicate minimum and maximumpressure, respectively.

properties at the inlet section are similar to the closed-mode case in that the entering flow hasa developed profile that is deflected by the stem. An essential difference with the closed-mode

case is that for the present open-mode case the deflection is in both upward and downwarddirection. The upward deflection causes fluid to relatively directly flow along stem and shafttowards the outlet section. The downward deflection leads to the formation of a recirculationzone in the lower-right corner of the valve chamber (Figure 5c) via which fluid diverges inlateral-upward direction in order to flow around the stem and eventually finds its way to theoutlet section as well (not shown). Fluid entering the outlet pipe is significantly acceleratedand directed. This leads to a flow that rapidly becomes developed. Lowering the temperatureresults, similar as for the closed-mode configuration, to slightly higher fluid velocities yet withretention of the qualitative flow structure.

Z

Y

X Z

Y

X

a) Velocity vectors. b) Velocity magnitude.

Figure 5: Full view of the velocity field in the symmetry plane for the open mode at m = 70µg/sand T = 265K. The velocity magnitude ranges from 0m/s (blue) to 3m/s (red).

The pressure field corresponding with the open-mode case considered above is given inFigure 7. Panel a gives the full view; panels b and c give close-ups of the outlet and inletsections, respectively. Both inlet and outlet sections have approximately linear pressure dropsdue to the developed nature of the flow. The most important feature is the formation ofa high-pressure zone at the right side of the stem, akin to that found for the closed-modeconfiguration, due to the sudden deflection of the flow. The pressure build-up leads, as before,to a local pressure that progressively exceeds the inlet pressure with increasing mass flow mand thus signifies a potential condensation (or freezing) zone. The risk of undesired phasechanges can again be averted by ensuring slightly superheated inlet conditions. The pressurefield for T = 245 is essentially the same yet with slightly amplified pressures.

Integral characterisation of the bypass valve

The FVM-analysis yields the following integral characterisation of the bypass valve:

• Incompressible flow in the laminar regime.

Z

Y

X Z

Y

X

a) Velocity vectors at inlet b) Velocity magnitude at inlet

Z

Y

X Z

Y

X

c) Velocity vectors at outlet d) Velocity magnitude at outlet

Figure 6: Close up of the velocity field in the symmetry plane for the open mode at m = 70µg/sand T = 265K. The velocity magnitude ranges from 0m/s (blue) to 3m/s (red).

• Isothermal conditions

• Pressure drop between inlet and outlet due to viscous friction. Correlation

∆p = A m2 + B, ∆p [Pa], m [µg/s], (2)

expresses the pressure drop ∆p as a function of the mass flow m, with

Z

Y

X

Z

Y

X Z

Y

X

a) Pressure in full view. b) Pressure at outlet. c) Pressure at inlet.

Figure 7: Pressure field in the symmetry plane in open mode at m = 70µg/s and T =265K. The pressure range is 110Pa; blue and red indicate minimum and maximum pressure,respectively.

T = 245; closed T = 245; closed T = 245; open T = 245; openA 0.016576 0.010302 0.021755 0.013172B 16.2306 10.2695 14.6568 9.0388

the corresponding coefficients A and B for the operating conditions considered above.Correlation (2) is graphically depicted in Figure 8a and clearly demonstrates the quadraticincrease in pressure drop with increasing mass flow. Generalisation to arbitrary inlettemperatures 245K ≤ T ≤ 265 is given below.

• Risk of condensation in case of saturation conditions at the inlet. Remedyagainst this unwanted effect is ensuring a slightly superheated state at the inlet. Thesuperheat must be higher for higher mass flows.

The changes in pressure drop appear to scale approximately linearly with changes in materialproperties. This follows readily from the following dimensional analysis. Steady laminar flowimplies that the balance between pressure gradient and viscous forces, i.e. ∇p = µ∇2u,predominates the fluid dynamics. Application to the above operating conditions gives

∆P ∼ µU

L→ ∆P265

∆P245=

µ265U265

µ245U245

U=m/Aρ−→ ∆P265

∆P245=

µ265ρ245

µ245ρ265=

ν265

ν245= 0.64, (3)

with ν = µ/ρ the kinematic viscosity and ∆P the order-of-magnitude estimate of the pressuredrop. Figure 8b reveals that the actual pressure-drop ratio ∆p265/∆p245 is in good agreementwith the estimate ∆P265/∆P245 = 0.64 throughout the operating range of the bypass valve.This suggests that within this range the pressure drop scales proportionally with the kinematicviscosity. Thus correlation (2) can to good approximation be generalised to

∆p =µ

µ245

[A245 m2 + B245

], (4)

with T = 245 the reference state with coefficients A and B as before and µ the kinematicviscosity corresponding with the inlet temperature of interest. Correlation (4) resides betweenthe correlations corresponding with T = 245K to T = 265K (Figure 4) and shifts smoothlyfrom the T = 245-curve to the T = 265-curve with increasing T .

0 20 40 60 80 100 1200

50

100

150

200

250

mass flow [µ g/s]

pres

sure

dro

p [P

a]

closed mode: T=245Kclosed mode: T=265K open mode: T=245K open mode: T=265K

30 40 50 60 70 80 90 100 1100

0.2

0.4

0.6

0.8

1

mass flow [µ g/s]

∆ P

265/∆

P245

closed modeopen mode

a) Closed mode. b) Open mode.

Figure 8: Total pressure drop ∆p between inlet and outlet as a function of the mass flow m(panel a) and ratios of pressure drop associated with minimum and maximum inlet tempera-tures (panel b). Symbols indicate numerical computations; curves in panel a are least-squaresfits for correlation (2); horizontal line in panel b indicates estimated pressure-drop ratio.

References

ALLPROPS software-package for evaluation of thermodynamic properties, University ofIdaho, College of Engineering, Center for Applied Thermodynamic Studies.

Bodendiek, F., Hollenbach, B. & Goncharov, K. 2005 Propylene loop heat pipe:technical note. Document AMS-OHB-TEN-003, OHB-System AG, Bremen.

Cengel, Y. A. & Boles, M. A. 2002 Thermodynamics. An Engineering Approach, Mc-Graw Hill (fourth edition), Boston.

Kundu, P. K. 1990 Fluid Mechanics, Academic Press, London.

Hirsch, C. 1988 Numerical Computation of Internal and External Flows, Wiley, Chichester.

Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory, Springer, Berlin.

Speetjens M. & Rindt, C. 2006 Analytical model for the bypass valve. INTERREG IIICMATEO-ANTASME Deliverable 9.1.

VDI Warmeatlas, Ninth Edition, 2002. Springer, Berlin.